Shaping and nonequiprobable signaling for intensity-modulated signals

Theory of shaping and nonequiprobable signaling, which has been developed for conventional electrical signals, must be modified to treat intensity-modulated (IM) signals. We show that for IM signals, the optimum shape of the constellation bounding region in N-dimensional (N-D) space is an N-D simplex. As N→∞, the maximum achievable shape gain is 1.33 dB (in terms of transmitted power), and the resulting marginal signaling distribution on the one-dimensional (1-D) constituent constellation is exponential. We also investigate the tradeoffs between shaping and its negative consequences, and find that a 1-dB shape gain can be achieved while incurring reasonable increases in peak-to-average power ratio and constellation expansion ratio